Advice for writing good mathematics?

Question

It's been a (far-fetched, possibly) goal of mine to some day write a math Textbook. I've been thinking about writing this question for a while, but reading an exceedingly mediocre text on Mathematical Modeling has finally provoked me. In several paragraphs, I have already spotted numerous changes I would have made to the book (not the least of which is publishing somewhere besides Elsevier, haha); so it got me thinking about mathematical exposition.

At this point I'd really just like writing advice. I've seen some articles e.g. (A Guide to Writing Mathematics and seen Serre's "Writing Mathematics Badly," and I know some of the "basic" authors that are celebrated expositors (Halmos, Spivak, Rudin...)

But I'm asking for you to share the benefits of your experience: Unforgivable mistakes you've seen made (repeatedly), great expositions you've read, or any other sort of advice you have (apart from 'just do it!' I am!).

Certainly I'm lucky to have this great website to practice on and see a vast number of others' style.

Lastly: I am not entirely sure if this question is appropriate, or if it is too broad or vague.

Answer 1

I've certainly not written a text, but I've read a few nasty ones. The best advice I would give an up-and-coming text writer:

1) Before you start writing, think long and hard about who you intend the book for -- Is it for teens? Undergraduates? Graduate Students? PhD wielders? This is something that really matters because of the "mathematical maturity" issue. Somebody that's been studying the subject for 20+ years will be better able to fill in any gaps you might leave (intentionally or unintentionally) in your treatment of the subject matter.

2) Super "symbol-heavy" books are not conducive to getting your message to your audience, although it might feed your ego. I think it is tempting for somebody who'd become so familiar with a subject that they can write a book about it to get carried away with showing off everything they know. If I had a quarter for every time I was reading something convoluted and came across the words 'trivially', 'obviously', 'clearly', etc... I would be a millionaire. Well, maybe not that rich, but I'd have at least $50.00.

3) It seems like mathematics texts come in one of two types: a) overly symbolic, cryptic, and dense; b) overly pedantic, example-ridden, boring to read.

4) Many mathematics texts seem to be very "method-driven" and not present the intrinsic subject matter they deal with. For instance, I'm currently reading a book on differential geometry that is extremely dense symbolically (granted, anything with tensors necessitates this to some degree) and the author hasn't bothered to tell my anything about why the hell what he is writing should matter to me: "Ok, I can do this curvature thing, and it's intrinsic, that's nice. So what do I do with it? What motivated the discussion in the first place?"

The above list is by no means complete, but I hope it helps a little. I encourage others to add to my list!

Disclaimer: I am partial to a more conversational reading style, as that is how I write. Don't be too formal in your writing style -- it's already mathematics for heaven's sake!

Best,

Dylan

Answer 2

I sympathize with some of the comments of Dylan Frank but I would argue to the OP that deciding who the audience is and the purpose of the book (point 1) affects everything to the point of somewhat invalidating some of the other complaints in Dylan Frank's answer: e.g. some books are supposed to be calculation and symbol heavy so that they act as efficient-if-somewhat-dry references for the people who already understand all the motivation and context but don't want to waste time rederiving things that are standard. Or some books are known for their myriad and challenging exercises and examples, amassed from thousands of pages of papers and other books. Sometimes, the advanced reader is thankful for a few `clearly' s because he/she knows all the standard arguments and just needs to skim a proof for one or two ideas. Obviously, for all of these things some books do them better than others, but in themselves I would not say that they are undesirable.

Basically, I think for the majority of mathematicians, the majority of textbooks fall short of what is being sought. However, this same fact means that for the majority of mathematicians there is at least one book which is more or less exactly what is being sought. So the diversity one sees in books of too wordy/too symbol-heavy or too dense/too drawn-out or too many examples and questions/no examples, just abstract theorems is a good thing; the literature is richer for it.

Having said that, obviously some books do manage to appeal to larger numbers than others and some are widely-regarded as excellent, but this is appropriately rare!

Answer 3

Motivation and examples are key.

For example, suppose you want to introduce the notion of a metric. Start a new section in your article/book. Begin the section by defining "distance in $\mathbb{R}^2$." Then show that this distance function satisfies the three properties that characterise a metric - but don't give the concept a name yet.

Then prove a bunch of stuff about distances in $\mathbb{R}^2$ using only those properties. Once the reader has gone, "Wow! Lots of theorems follow from just those three facts about distance," you should invite them to ask themselves "well, what DOESN'T follow from just those three properties???"

The time is ripe to define the terms "metric" and "metric space," and to point out that all the theorems proved so far are valid for arbitrary metric spaces. The next step is to construct examples of metric spaces where a variety of intuitively sensible theorems fail. For each such example, work out what additional assumptions $\mathbb{R}^2$ satisfies, that an arbitrary metric space may fail to satisfy, in order for the theorems to hold. Each theorem that holds for $\mathbb{R}^2$, but fails for some metric space, results in a motivation for a definition.

Now so far, all examples of metrics, except for the one on $\mathbb{R}^2$, have been pathological cases. At some point, however, you also want to start a new section, and begin by introducing examples of metrics/metric spaces that are actually useful. This provides the final justification for the terminology that has been introduced, and by this point you can safely assume the reader has been thoroughly initiated into the study of general metric spaces.

Notice also that you started the both sections with examples. That's good - it means you can wash, rinse and repeat.

Edit: Keep this metaphor in mind. Your purpose is not to teach the reader anything. Its to bring out their natural curiosity in the subject matter. Thus, you should set things up so that the reader is already yearning for a concept (or something very much like it) before you introduce it.

Answer 4

There are at least two cognitive processes happen during the information transmission:

• Illusion of transparency or curse of knowledge: the author always overestimates how the readers understand their work. In an experiment about guessing a song by listening to how it's tapped, the tappers guess that 50% of time the listeners will guess right, while in fact the number is 3%.
• Cognitive load: the readers read the paper to find solutions for their problems, not the author's problem. What is worse is that the desire to get them solved will cloud their minds and make what is written black and white in the paper be unavoidably exotic, misunderstood or, to an extent, distorted. Not getting what you are expecting is so frustrating, like you are in a rush and your car suddenly stops working. A related phenomenon is information overload.

Of course it is impossible to write a smooth research paper, and the readers are expected to overcome it anyway, and I may exaggerate a bit, but overall it's still better to keep them in mind when writing anything. It's important to have a concise writing, but when the work is long, complicate and abstruse like in a math paper, only relying on conciseness may make it become too dense, and will backfire the intention.

My advice are:

• Make a concrete analogy. A concrete analogy will intertwine to the text and allow room for the readers to project their background into it.
• Make the ideas constantly contradict each other. "Contradiction" here doesn't mean as a logical contradiction, but more about "a surprising, but still logical step of development". It introduces why the topic is important, and is the source of excitation, enlightenment, and satisfaction. Being able to solve contradictions is the reason why the ideas survive and are worth the attention.
• Notice where the flow emerges and dissipates. This will help overcome the jargon barrier without having to oversimplify them. Imagine the article is like a heatmap, and each jargon/theorem/proof is a heat source, then the writer's job is to locate them not too hot (too dense) or too cold (too uninformative).
• Viewing the topic in as much perspective as possible. Each unit of the article (phrase, sentence, paragraph, section) should be thought as an unique, different perspective, so the readers can see the topic in fresh perspectives. This makes a profound, advanced topic more playful, imaginative and transformative. The simplest trick is to start a new sentence with a different subject than the preceding one. If the readers are about to leave and we have only one more sentence to keep their attention, what should it be?

Though not about math paper, research about myth debunking and web usability (widely known for the F-shaped pattern) will provide experimental insights about what conciseness really means. For complicated and long writings, I also have an article for this, you can check it out: Making concrete analogies and big pictures.

Answer 5

Just like any sort of writing what makes one's work remarkable is uniqueness and originality in it. If you want your work to be remembered for some good amount of time, then make sure to include hard problem , to give students a hard time.(the questions must also be original) a very good example of such a book is Problems in physics by IE Irodov (though its about physics, but I would love if a similar book for maths also existed)

If you want your work to be spread to a lot of students, then make sure you have a lot of easy problems.

what i particularly hate in most maths text books is that there are a lot a questions , a hell lot of them , and most of them are repetitive and irrelevant, so you might want to take care of that while writing your book.